Retriangular numbers and affine maps between polygonal numbers

Source abstract

A positive integer $a$ is called \emph{retriangular} if there is a positive integer $b$ such that $aT+b$ is a triangular number whenever $T$ is. We show that the retriangular numbers are exactly the odd squares greater than $1$, and that for each of them $b$ is unique. The proof rests on a rigidity principle: a non-constant affine map carrying the values of one quadratic sequence into the values of another is induced by an affine map of the plane sending the vertex of one parabola to the vertex of the other. We use it to classify all affine maps between sets of polygonal numbers; the $s$-gonal numbers map affinely into the $t$-gonal numbers if and only if $M_t\mid M_s$, where $M_s=2(s-2)/\gcd(s,4)$ defines a bijection from $\{3,4,5,\dots\}$ onto the positive integers. We also treat sparse data, polynomial maps, figurate numbers of higher degree and sums of two triangular numbers, and we study the monoid of affine self-maps of an arbitrary set of integers. For the $s$-gonal numbers this monoid is an arithmetical congruence monoid, and the factorization of self-maps into irreducible ones is unique only for triangular and square numbers.